Percolation model

Like SAWs that are related to the 0(m)-model, percolation is connected to the q-state Potts model. The g-state Potts model [86] is a spin model whose high temperature expansion (when g 1) corresponds to diagr lms that are percolation configurations. In the Potts model to the each site a of a regulcu lattice a spin variable Ox corresponds which can be in q possible states (g = 2 corresponds to the Ising case). Interactions favor nearest neighbors which are in the same state. The effective Hamiltonian for the model reads ... 

Figure 7.5. The analogy among three different phenomena ferromagnetic ordering of lattice spins in clusters of size up to as calculated according to the Ising model percolation clusters of size smaller than and diffusion-limited growth with size fluctuations up to All have the same power law behavior v is the reciprocal fractal dimension, p is the occupation fraction, and M the mass. From H. E. Stanley. In Random Fluctuations and Pattern Growth, p. 1. With kind permission from Kluwer Academic Publishers.

Bauhofer W, kovacs JZ (2009) A review and analysis of electrical percolation in carbon nanotube polymer composites. Compos Sci Technol 69 1486 Behnam A, Guo J, Ural A (2007) Effects of nanotube alignment and measurement direction on percolation resistivity in single-walled carbon nanotube films. J Appl Phys 102 044313 Berhan L, Sastry SM (2007) Modeling percolation in high-aspect-ratio fiber systems. L Soft-core versus hard-core models. Phys Rev E 75 041120 Berman D, Orr BG, Jaeger HM, Goldman AM (1986) Conductances of filled two-dimensional networks. Phys Rev B 33 4301... 

More recently, different approaches (cellular models, percolation analogy, fractal structure, blobs and links) have been proposed, to account for the porous volume and calculate the evolution of the meehanical properties as a function of the structural characteristics. Such models seem attraetive to deseribe the meehanieal properties of gels for several reasons. In eontrast with the empirieal relationships, they try to relate the physieal properties to a description of the mean strueture, or to the aggregation process they also predict that the Poisson ratio is with the fraetion of the solid phase, which is an experimental result demonstrated for aerogels and PDA. Another interesting feature of those models is that they prediet a power law evolution of the meehanieal properties as a funetion of the... 

Berhan, L. and Sastry, A.M. (2007) Modeling percolation in nanotube reinforced composite materials part one - soft-core versus hard-core models. Phys. Rev. E, 75, 041120-1-041120-8. 

For completeness, note that several researchers have exploited the well-developed analytical theories of the stmcture of fluids to model percolation in mixtures of interacting particles. By proposing various extensions of the multicomponent Omstein-Zernike equation, coupled with connectivity definitions from continuum percolation theories, simplified analytical expressions are derived for the percolation threshold of a composite system subjected to interparticle and medium-induced interactions. However, to date, simulations dominate the study of dynamic percolation. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

The percolation argument is based on the idea that with an increasing Cr content an insoluble interlinked cliromium oxide network can fonn which is also protective by embedding the otherwise soluble iron oxide species. As the tlireshold composition for a high stability of the oxide film is strongly influenced by solution chemistry and is different for different dissolution reactions [73], a comprehensive model, however, cannot be based solely on geometrical considerations but has in addition to consider the dissolution chemistry in a concrete way. 

We examined the role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission, as shown in Fig. 11 [1,2]. In this experiment, a metal net of modulus Eo containing No = 10" bonds was stressed and held at constant strain (ca. 2%) on a tensile tester. A computer randomly selected a bond, which was manually cut, and the relaxation of the net modulus was measured. The initial relaxation process as a function of the number of bonds cut N, could be well described by the effective medium theory (EMT) via... 

Fig. I I, The role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission.

By placing diblocks or random copolymers of aerial density X, at incompatible interfaces (Fig. 16), following the work of Creton, Brown and Kramer et al. [59,60,81-83], the percolation model predicts that XL... 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

M. Kolb, Y. Boudeville. Kinetic model for heterogeneous catalysis Cluster and percolation properties. J Chem Phys 92 3935-3945, 1990. 

Composilion has a marked effect on p. Dilution can cause (i to drop by orders of magnitude. The functional dependence is often expressed in terms of an exponential dependence on intersite distance R=at m as suggested by the homogeneous lattice gas model, p c)exponential distance dependence of intersite coupling precludes observing a percolation threshold for transport. 

Here, is an effective overlap parameter that characterizes the tunneling of chaiges from one site to the other (it has the same meaning as a in Eq. (14.60)). T0 is the characteristic temperature of the exponential distribution and a0 and Be are adjustable parameters connected to the percolation theory. Bc is the critical number of bonds reached at percolation onset. For a three-dimensional amorphous system, Bc rs 2.8. Note that the model predicts a power law dependence of the mobility with gate voltage. 

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

---